The Fluid Limit of a Shortest Remaining Processing Time Queue.

Speaker: 

Professor Amber Puha

Institution: 

Cal State San Marcos

Time: 

Saturday, December 1, 2007 - 2:15pm

Location: 

McDonnell Douglas Auditorium

Consider a GI/GI/1 queue operating under shortest remaining processing time with preemption. To describe the evolution of this system, we use a measure valued process that keeps track of the residual service times of all jobs in the system at any given time. Of particular interest is the waiting time for large jobs, which can be tracked using the frontier process, the largest service time of any job that has ever been in service. We propose a fluid model and present a functional limit theorem justifying it as an approximation of this system. The fluid model state descriptor is a measure valued function for which the left edge of the support is the fluid analog for the frontier process.
Under mild assumptions, we prove existence and uniqueness of fluid model solutions.
Furthermore, we are able to characterize the left edge of fluid model
solutions as the right continuous inverse of a simple functional of the initial condition,
arrival rate, and service time distribution. When applied to various examples, this
characterization reveals the dependence on service time distribution of the rate at which the
left edge of the fluid model increases.

On Volatilities

Speaker: 

Professor Jean-Pierre Fouque

Institution: 

UCSB

Time: 

Saturday, December 1, 2007 - 1:15pm

Location: 

McDonnell Douglas Auditorium

The various concepts of volatility (realized, local, stochastic, implied), well defined or depending on a given model and/or statistical estimates, will be discussed. Backward and forward equations for call-option payoffs (Black-Scholes and Dupire equations) will be revisited. We will show that, besides the Black-Scholes model with constant volatility, fast mean reverting stochastic volatility models can reconcile local and implied volatilities. If time permits we will also look at the relation between volatility and correlation in the multidimensional case.
The talk is addressed to a general audience in Probability without any particular deep background in financial mathematics.

Large time fluctuations of the totally asymmetric simple exclusion process.

Speaker: 

Professor Alexei Borodin

Institution: 

Cal Tech

Time: 

Saturday, December 1, 2007 - 11:00am

Location: 

McDonnell Douglas Auditorium

The totally asymmetric simple exclusion process (TASEP) is one of the
simplest models of interacting particle systems on the one-dimensional
lattice. It is equivalent to a random growth model from the
Kardar-Parisi-Zhang universality class. We focus on fluctuations of the
particle positions for a nonequilibrium TASEP that starts from certain
deterministic initial conditions. We (rigorously) derive the scaling
exponents 1/3 and 2/3, and identify the limit laws as those of Gaussian
Orthogonal and Unitary ensembles of the random matrix theory.

The effect of disorder on polymer depinning transitions.

Speaker: 

Professor Ken Alexander

Institution: 

USC

Time: 

Saturday, December 1, 2007 - 10:00am

Location: 

McDonnell Douglas Auditorium

We consider a directed polymer pinned by one-dimensional quenched randomness, modeled by the space-time trajectories of an underlying Markov chain which encounters a random potential of form u + V_i when it visits a particular site, denoted 0, at time i. The polymer depins from the potential when u goes below a critical value. We consider in particular the case in which the excursion length (from 0) of the underlying Markov chain has power law tails. We show that for certain tail exponents, for small inverse temperature \beta there is a constant D(\beta), approaching 0 with \beta, such that if the increment of u above the annealed critical point is a large multiple of D(\beta) then the quenched and annealed systems have very similar free energies, and are both pinned, but if the increment is a small multiple of D(\beta), the annealed system is pinned while the quenched is not. In other words, the breakdown of the ability of the quenched system to mimic the annealed occurs entirely at order D(\beta) above the annealed critical point.

Random sampling and probability

Speaker: 

Professor Richard Bass

Institution: 

University of Connecticutt

Time: 

Tuesday, November 20, 2007 - 11:00am

Location: 

MSTB 254

The ``random sampling'' in the title has nothing whatsoever to
do with statistics. Instead, it refers to the perfect reconstruction of
a band-limited function from samples, a classical problem of
Fourier analysis and signal processing. With deterministic sampling,
almost everything is known in one dimension and almost nothing
is known in higher dimensions. It turns out that one loses very little in
efficiency by using random sampling, and in return one can use
probabilistic techniques to get some interesting theoretical results.
This is joint work with Karlheinz Gr\"ochenig.

Martingale Functions of Brownian Motion and Its Local Time at 0

Speaker: 

Professor Patrick Fitzsimmons

Institution: 

UCSD

Time: 

Tuesday, November 6, 2007 - 11:00am

Location: 

MSTB 254

Let $B = (B_t: t\ge 0)$ be a real-valued Brownian motion and let
$L = (L_t: t\ge 0)$ denote its local time in state 0. We present a characterization of the measurable functions $H$ such that $M_t = H(B_t,L_t)$
is a continuous local martingale. It turns out that the class of such functions is considerably wider when one relaxes the smoothness conditions that would be needed for a facile application of It\^o's formula.

"Trace processes and global limit theorems"

Speaker: 

Professor Nicola Squartini

Institution: 

UNC Charlotte

Time: 

Tuesday, October 23, 2007 - 11:00am

Location: 

MSTB 254

We collect together a number of examples of random walk
where the characteristic function of the first step has a
singularity at the point t=0. The function \log\varphi(t) has
two different expansions for positive and negative $t$ near the
origin; we call the coefficients of these expansions left and
right quasicumulants. Such examples include the trace of a
two dimensional random walk {(X_n,Y_n)} on the x-axis, and the
subordinated random walk (X_{\tau_n}) where (\tau_n) is an
appropriate sequence of random times. Using quasicumulants we derive an asymptotic expansion for the distribution of the sums of i.i.d. random variables, and assuming
further differentiability condition we are able to give sharp
estimate in the variable x of the remainder term.

Multi-frequency imaging of multiple targets in Rician fading media: stability and resolution

Speaker: 

Professor Mike Yan

Institution: 

UC Davis

Time: 

Tuesday, November 13, 2007 - 9:00am

Location: 

MSTB 254

Imaging of obscured targets in random media is a difficult and important problem. One of the central questions is that of stability which is particularly relevant to imaging in stochastic media. The main goal of the research is to develop a general criterion for multiple-frequency array imaging of multiple targets in stochastic media. An important feature of the cluttered media we considered here is that the coherent or mean signals do not vanish. It is called Rician fading medium in communication. Foldy-Lax formulas are used to simulate the exact wave propagations in the random media. In the talk, I will propose two models: passive array model and active array model. Then the stabilities of the imaging function with multiple point targets are given in both cases, followed by the numerical simulations to show the consistency with the analysis. If time permits, I will give some preliminary results about the stability conditions of the multiple extended targets, as well as the simulations.

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